Subpixel-based image down-sampling

ABSTRACT

Systems, methods, and apparatus for sampling images using minimum mean square error subpixel-based down-sampling (MMSE-SD) are presented herein. A partition component can receive a first array of pixels, and divide the first array of pixels into two-dimensional (2-D) blocks of pixels. Further, a sampling component can diagonally down-sample subpixels of a block of the 2-D blocks, and generate a second array of pixels based on the down-sampled subpixels. The sampling component can alternately sample subpixels of adjacent pixels of the block in a diagonal direction, and generate the second array of pixels based on the subpixels. A reconstruction component can create a virtual image based on, at least in part, the second array of pixels. A MMSE-SD component can determine an optimal low resolution image based on, at least in part, respective color components of the virtual image and a high resolution image associated with the first array of pixels.

PRIORITY CLAIM

This application claims priority to U.S. Provisional Patent ApplicationSer. No. 61/282,620, filed on Mar. 9, 2010, entitled “NOVEL 2-D MMSESUBPIXEL-BASED IMAGE DOWN-SAMPLING.” The entirety of the aforementionedapplication is incorporated by reference herein.

TECHNICAL FIELD

This disclosure relates generally to image processing including, but notlimited to, minimum mean square error subpixel-based down-sampling(MMSE-SD).

BACKGROUND

With the advance of portable technologies, down-sampling of highresolution image information is often required to display highresolution images(s) and/or video(s), e.g., high-definition (HD)television (HDTV) information, HD movies, etc. on a lower resolutiondisplay, e.g., included in a handheld device such as a cellular phones,a portable multimedia player (PMP), a personal data assistant (PDA),etc.

A color pixel of a high resolution matrix display, e.g. liquid crystaldisplay (LCD), plasma display panel (PDP), etc. includes threesubpixels, each subpixel representing one of three primary colors, i.e.,red (R), green (G), and blue (B). Although the subpixels are notseparately visible, they are perceived together as color(s). Oneconventional technique for down-sampling a high resolution, e.g., color,image is pixel-based down-sampling, which selects every third pixel ofthe high resolution image to display. Such down-sampling severelyaffects shapes and/or details of the image, as over 30% of informationof the image is compressed (or lost). Further, pixel-based down-samplingcauses aliasing, or distortion, of the image near shape edges.

Another conventional technique for down-sampling a high resolution imageis subpixel-based down-sampling, which alternately selects red, green,and blue subpixels from consecutive pixels of a block of pixels of thehigh resolution image in a horizontal direction. As such, the (i,j)pixel in the downsampled image includes subpixels (R_(i,j), G_(i,j+1),B_(i,j+2)) of the block of pixels—the subscripts denoting pixel indicesof the block of pixels. Although such subpixel-based down-samplingpreserves the shapes of images more effectively than pixel-baseddown-sampling, resulting subpixel-based images incur more colorfringing, i.e., artifacts, around non-horizontal edges than pixel-baseddownsampled images.

The above-described deficiencies of today's image down-samplingtechniques and related technologies are merely intended to provide anoverview of some of the problems of conventional technology, and are notintended to be exhaustive. Other problems with the state of the art, andcorresponding benefits of some of the various non-limiting embodimentsdescribed herein, may become further apparent upon review of thefollowing detailed description.

SUMMARY

The following presents a simplified summary to provide a basicunderstanding of some aspects described herein. This summary is not anextensive overview of the disclosed subject matter. It is not intendedto identify key or critical elements of the disclosed subject matter, ordelineate the scope of the subject disclosure. Its sole purpose is topresent some concepts of the disclosed subject matter in a simplifiedform as a prelude to the more detailed description presented later.

To correct for the above identified deficiencies of today's imageprocessing environments and other drawbacks of conventional imagedown-sampling environments, various systems, methods, and apparatusdescribed herein sample images using MMSE-SD.

For example, a method can include partitioning a two-dimensional (2-D)array of pixels into 2-D blocks of pixels; alternately samplingsubpixels of a block of pixels of the 2-D blocks of pixels in a diagonaldirection; and generating an image based on a result of the alternatelysampling subpixels of the block of pixels. Further, the method caninclude deriving, based on the image, a virtual image according to asize of an other image associated with the 2-D array of pixels. Thegenerating the image can include minimizing a mean square error betweenthe virtual image and the other image; and determining at least onecolor component of a low resolution image associated with the resultbased on a block circulant matrix.

In another example, a system can include a partition componentconfigured to receive a first array of pixels; and divide the firstarray of pixels into two-dimensional (2-D) blocks of pixels. Further,the system can include a sampling component configured to diagonallydown-sample subpixels of a block of the 2-D blocks; and generate asecond array of pixels based on the down-sampled subpixels. Furthermore,the sampling component can alternately sample a red subpixel, a greensubpixel, and a blue subpixel of adjacent pixels of the block in adiagonal direction; and generate the second array of pixels based on thered subpixel, the green subpixel, and the blue subpixel.

Moreover, the system can include a reconstruction component configuredto create a virtual image based on, at least in part, the second arrayof pixels. The reconstruction component can create the virtual imageutilizing a directional weighted average of neighboring subpixels of thesecond array of pixels. Further, the system can include a minimum meansquare error (MMSE) subpixel-based down-sampling (MMSE-SD) componentconfigured to determine an optimal low resolution image based on, atleast in part, respective color components of the virtual image and ahigh resolution image associated with the first array of pixels. TheMMSE-SD component can minimize a mean square error between the virtualimage and the high resolution image; and determine the optimal lowresolution image based on the mean square error.

In yet another example, an apparatus can include means for dividing ahigh resolution image into blocks of pixels; means for selectingsubpixels of a block of the blocks in a diagonal direction; and meansfor creating a low resolution image based on the selected subpixels.Further, the apparatus can include means for creating a virtual highresolution image using the low resolution image; means for minimizing amean square error between the high resolution image and the virtualimage; and means for determining an optimal low resolution image basedon an output of the means for minimizing the mean square error.

The following description and the annexed drawings set forth in detailcertain illustrative aspects of the disclosed subject matter. Theseaspects are indicative, however, of but a few of the various ways inwhich the principles of the innovation may be employed. The disclosedsubject matter is intended to include all such aspects and theirequivalents. Other advantages and distinctive features of the disclosedsubject matter will become apparent from the following detaileddescription of the innovation when considered in conjunction with thedrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Non-limiting and non-exhaustive embodiments of the subject disclosureare described with reference to the following figures, wherein likereference numerals refer to like parts throughout the various viewsunless otherwise specified.

FIG. 1 illustrates a block diagram of a subpixel-based down-samplingsystem, in accordance with an embodiment.

FIG. 2 illustrates a block diagram of a two-dimensional high resolutionimage, in accordance with an embodiment.

FIG. 3 illustrates a block diagram of a pixel, in accordance with anembodiment.

FIG. 4 illustrates a block diagram of a subpixel-based down-samplingmodel, in accordance with an embodiment.

FIG. 5 illustrates an example of down-sampling a block of pixels, inaccordance with an embodiment.

FIG. 6 illustrates another example of down-sampling a block of pixels,in accordance with an embodiment.

FIG. 7 illustrates a block diagram of a system for reconstructing a highresolution image, in accordance with an embodiment.

FIG. 8 illustrates a block diagram of a subpixel-based reconstructionmodel, in accordance with an embodiment.

FIG. 9 illustrates a block diagram of a minimum mean square errordown-sampling system, in accordance with an embodiment.

FIG. 10 illustrates a block diagram of a down-sampling environmentincluding a display, in accordance with an embodiment.

FIGS. 11-12 illustrate various processes associated with minimum meansquare error subpixel-based down-sampling (MMSE-SD), in accordance withan embodiment.

FIG. 13 illustrates a block diagram of a computing system operable toexecute the disclosed systems and methods, in accordance with anembodiment.

DETAILED DESCRIPTION

Various non-limiting embodiments of systems, methods, and apparatuspresented herein sample images using minimum mean square errorsubpixel-based down-sampling (MMSE-SD).

In the following description, numerous specific details are set forth toprovide a thorough understanding of the embodiments. One skilled in therelevant art will recognize, however, that the techniques describedherein can be practiced without one or more of the specific details, orwith other methods, components, materials, etc. In other instances,well-known structures, materials, or operations are not shown ordescribed in detail to avoid obscuring certain aspects.

Reference throughout this specification to “one embodiment,” or “anembodiment,” means that a particular feature, structure, orcharacteristic described in connection with the embodiment is includedin at least one embodiment. Thus, the appearances of the phrase “in oneembodiment,” or “in an embodiment,” in various places throughout thisspecification are not necessarily all referring to the same embodiment.Furthermore, the particular features, structures, or characteristics maybe combined in any suitable manner in one or more embodiments.

As utilized herein, terms “component,” “system,” “interface,” and thelike are intended to refer to a computer-related entity, hardware,software (e.g., in execution), and/or firmware. For example, a componentcan be a processor, a process running on a processor, an object, anexecutable, a program, a storage device, and/or a computer. By way ofillustration, an application running on a server and the server can be acomponent. One or more components can reside within a process, and acomponent can be localized on one computer and/or distributed betweentwo or more computers.

Further, these components can execute from various computer readablemedia having various data structures stored thereon. The components cancommunicate via local and/or remote processes such as in accordance witha signal having one or more data packets (e.g., data from one componentinteracting with another component in a local system, distributedsystem, and/or across a network, e.g., the Internet, a local areanetwork, a wide area network, etc. with other systems via the signal).

As another example, a component can be an apparatus with specificfunctionality provided by mechanical parts operated by electric orelectronic circuitry; the electric or electronic circuitry can beoperated by a software application or a firmware application executed byone or more processors; the one or more processors can be internal orexternal to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts; the electroniccomponents can include one or more processors therein to executesoftware and/or firmware that confer(s), at least in part, thefunctionality of the electronic components. In an aspect, a componentcan emulate an electronic component via a virtual machine, e.g., withina cloud computing system.

The word “exemplary” and/or “demonstrative” is used herein to meanserving as an example, instance, or illustration. For the avoidance ofdoubt, the subject matter disclosed herein is not limited by suchexamples. In addition, any aspect or design described herein as“exemplary” and/or “demonstrative” is not necessarily to be construed aspreferred or advantageous over other aspects or designs, nor is it meantto preclude equivalent exemplary structures and techniques known tothose of ordinary skill in the art. Furthermore, to the extent that theterms “includes,” “has,” “contains,” and other similar words are used ineither the detailed description or the claims, such terms are intendedto be inclusive—in a manner similar to the term “comprising” as an opentransition word—without precluding any additional or other elements.

Artificial intelligence based systems, e.g., utilizing explicitly and/orimplicitly trained classifiers, can be employed in connection withperforming inference and/or probabilistic determinations and/orstatistical-based determinations as in accordance with one or moreaspects of the disclosed subject matter as described herein. Forexample, an artificial intelligence system can be used to automaticallypartition, e.g., via partition component 110, a 2-D array of pixels into2-D blocks of pixels. Further, the artificial intelligence system can beused to automatically alternately sample, e.g., via sampling component120, subpixels of a block of pixels of the 2-D blocks of pixels in adiagonal direction; and generate an image based on such sampling.Furthermore, the artificial intelligence system can derive, e.g., viareconstruction component 710, a virtual image according to a size ofanother image associated with the 2-D array of pixels.

As used herein, the term “infer” or “inference” refers generally to theprocess of reasoning about, or inferring states of, the system,environment, user, and/or intent from a set of observations as capturedvia events and/or data. Captured data and events can include user data,device data, environment data, data from sensors, sensor data,application data, implicit data, explicit data, etc. Inference can beemployed to identify a specific context or action, or can generate aprobability distribution over states of interest based on aconsideration of data and events, for example.

Inference can also refer to techniques employed for composinghigher-level events from a set of events and/or data. Such inferenceresults in the construction of new events or actions from a set ofobserved events and/or stored event data, whether the events arecorrelated in close temporal proximity, and whether the events and datacome from one or several event and data sources. Various classificationschemes and/or systems (e.g., support vector machines, neural networks,expert systems, Bayesian belief networks, fuzzy logic, and data fusionengines) can be employed in connection with performing automatic and/orinferred action in connection with the disclosed subject matter.

In addition, the disclosed subject matter can be implemented as amethod, apparatus, or article of manufacture using standard programmingand/or engineering techniques to produce software, firmware, hardware,or any combination thereof to control a computer to implement thedisclosed subject matter. The term “article of manufacture” as usedherein is intended to encompass a computer program accessible from anycomputer-readable device, computer-readable carrier, orcomputer-readable media. For example, computer-readable media caninclude, but are not limited to, a magnetic storage device, e.g., harddisk; floppy disk; magnetic strip(s); an optical disk (e.g., compactdisk (CD), a digital video disc (DVD), a Blu-ray Disc™ (BD)); a smartcard; a flash memory device (e.g., card, stick, key drive); and/or avirtual device that emulates a storage device and/or any of the abovecomputer-readable media.

Conventional downsampling techniques negatively affect shapes and/ordetails of a sampled image, causing aliasing of the sampled image nearshape edges, and/or causing increased color fringing aroundnon-horizontal edges of the sampled image. Compared to such technology,various systems, methods, and apparatus described herein in variousembodiments can improve sampling of images by using MMSE-SD.

Referring now to FIG. 1, a block diagram of a subpixel-baseddown-sampling system 100 is illustrated, in accordance with anembodiment. Aspects of system 100, and systems, networks, otherapparatus, and processes explained herein can constitutemachine-executable instructions embodied within machine(s), e.g.,embodied in one or more computer readable mediums (or media) associatedwith one or more machines. Such instructions, when executed by the oneor more machines, e.g., computer(s), computing device(s), virtualmachine(s), etc. can cause the machine(s) to perform the operationsdescribed.

Additionally, the systems and processes explained herein can be embodiedwithin hardware, such as an application specific integrated circuit(ASIC) or the like. Further, the order in which some or all of theprocess blocks appear in each process should not be deemed limiting.Rather, it should be understood by a person of ordinary skill in the arthaving the benefit of the instant disclosure that some of the processblocks can be executed in a variety of orders not illustrated.

As illustrated by FIG. 1, system 100 can include partition component 110and sampling component 120. As described above, down-sampling is aprocedure that can be used to display high resolution images/video vialower resolution devices. In an aspect, partition component 110 canreceive a high resolution image (L) including a two-dimensional (2-D)array of pixels. Further, partition component can partition, divide,etc. the 2-D array of pixels into 2-D blocks of pixels, e.g., intoblocks including a 3×3 array of pixels. Sampling component 120 candown-sample each block, or 3×3 array of pixels, by selecting, sampling,etc. subpixels of pixels of the block in a diagonal direction togenerate a low resolution image (S).

FIG. 2 illustrates a block diagram of a 2-D high resolution image (L)200 including pixels 210, in accordance with an embodiment. Pixels 210are addressable screen elements of a display, arranged in a 2-D grid.Each pixel 210 is addressed by coordinates (not shown), which can bearbitrarily assigned and/or re-assigned during image processing. Asillustrated by FIG. 3, pixel 210 can include three subpixels: redsubpixel 310, green subpixel 320, and blue subpixel 330. Subpixels 310,320, and 330, which together represent color when perceived at adistance, are also addressed by coordinates.

Referring now to FIG. 2, partition component 110 can be configured todivide high resolution image 200 into at least two blocks 205 and 215that include a 3×3 array of pixels 210. In one embodiment, highresolution image 200 can include 3M×2N pixels. Further, in anotheraspect illustrated by FIG. 4, partition component 110 can associatecoordinates (i,j) with pixels 210 of blocks 205 and 215. As such, asillustrated by FIG. 5, sampling component 120 can be configured todiagonally select, copy, sample, etc. red subpixel 310 at coordinateR_(3i−2,3j−2), green subpixel 320 at coordinate G_(3i−1,3j−1), and bluesubpixel 330 at coordinate B_(3i,3j) from pixels 210 of blocks 205/215in a first diagonal direction 510 to create sample 520 corresponding toa pixel of the low resolution, down-sampled, image S (see FIG. 1). Assuch, down-sampled, low-resolution image S can include M×N pixelscorresponding to the 3M×2N pixels of high resolution image 200.

In another embodiment illustrated by FIG. 6, sampling component 120 canbe configured to diagonally select, copy, sample, etc. red subpixel 310at coordinate R_(3i,3j−2), green subpixel 320 at coordinateG_(3i−1,3j−1), and blue subpixel 330 at coordinate B_(3i−2,3j) frompixels 210 of blocks 205/215 in a second diagonal direction 610 tocreate sample 620 corresponding to a (i,j)^(th) pixel (r_(i,j), g_(i,j),b_(i,j)) of the low resolution, down-sampled image S. As such, system100 can more effectively preserve shape details of a high resolutionimage than conventional down-sampling techniques.

Now referring to FIG. 7, a block diagram of a system 700 forreconstructing a high resolution image L is illustrated, in accordancewith an embodiment. System 700 can include a reconstruction component710 that can be configured to receive a low resolution image, e.g., lowresolution image S generated by sampling system 100. In an aspect,reconstruction component 710 can be configured to derive a virtual image(L′), approximating high resolution image L, based on, at least in part,a reconstruction model 800 for red component (or subpixel) generationillustrated by FIG. 8. As illustrated by 805, each square ofreconstruction model 800 represents a pixel in virtual image L′, inwhich α_(k), β_(k), k=1, 2, 3, 4; and α_(k)+β_(k)=1 for each k issatisfied.

As such, the (i,j)^(th) pixel (r_(i,j), g_(i,j), b_(i,j)) in Scorresponds to a 3×3 block, or array 810, of pixels in L′ at locations(k, l), with k=3i−2, 3i−1, or 3i; and l=3j−2, 3j−1, or 3j.Reconstruction component 710 can copy red subpixel r_(i,j) of the(i,j)^(th) pixel of S to a first location (3i−2, 3j−2), at 820, of L′;green subpixel g_(i,j) of the (i, j)^(th) pixel of S to a secondlocation (3i−1, 3j−1), at 830, of L′; and blue subpixel b_(i,j) of the(i, j)^(th) pixel of S to a third location (3i, 3j), at 840, of L′.Further, reconstruction component 710 can generate red components (orred subpixels) neighboring the first location using a directionalweighted average.

For example, reconstruction component 710 can generate red componentsmissing from locations (3i−3, 3j−3) of L′ and (3i−4, 3j−4) of L′ basedon equations (1) and (2), respectively, as follows:

α₃r_(i,j)+β₃r_(i−1,j−1),  (1)

β₃r_(i,j)+α₃r_(i−1,j−1).  (2)

Further, reconstruction component 710 can generate red componentsmissing from locations (3i−2, 3j−1) of L′ and (3i−2, 3j) of L′ based onequations (3) and (4), respectively, as follows:

α₁r_(i,j)+β₁r_(i,j+1),  (3)

β₁r_(i,j)+α₁r_(i,j+1).  (4)

In another aspect, reconstruction component 710 can generate greencomponents missing from locations (3i−2, 3j−2) of L′ and (3i−3, 3j−3) ofL′ based on equations (5) and (6), respectively, as follows:

α₃g_(i,j)+β₃g_(i−1,j−1),  (5)

β₃g_(i,j)+α₃g_(i−1,j−1).  (6)

Further, reconstruction component 710 can generate green componentsmissing from locations (3i−1, 3j) of L′ and (3i−1, 3j+1) of L′ based onequations (7) and (8), respectively, as follows:

α₁g_(i,j)+β₁g_(i,j+1),  (7)

β₁g_(i,j)+α₁g_(i,j+1).  (8)

In yet another aspect, reconstruction component 710 can generate bluecomponents missing from locations (3i−1, 3j−1) of L′ and (3i−2, 3j−2) ofL′ based on equations (9) and (10), respectively, as follows:

α₃b_(i,j)+β₃b_(i−1,j−1),  (9)

β₃b_(i,j)+α₃b_(i−1,j−1).  (10)

Further, reconstruction component 710 can generate blue componentsmissing from locations (3i, 3j+1) of L′ and (3i, 3j+2) of L′ based onequations (11) and (12), respectively, as follows:

α₁b_(i,j)+β₁b_(i,j+1),  (11)

β₁b_(i,j)+α₁b_(i,j+1).  (12)

In one aspect, reconstruction component 710 can copy red subpixelr_(i−1,j−1) of the (i−1, j−1)^(th) pixel of S to location (3i−5, 3j−5),at 850, of L′; green subpixel g_(i−1,j−1) of the (i−1, j−1)^(th) pixelof S to a second location (3i−4, 3j−4), at 860, of L′; and blue subpixelb_(i−1,j−1) of the (i−1, j)^(th) pixel of S to a third location (3i,3j), at 870, of L′. Further, reconstruction component 710 can generatered components (or red subpixels) neighboring r_(i−1,j−1) using adirectional weighted average of respective neighboring components asdescribed above.

Further, reconstruction component 710 can copy red subpixels r_(i−1,j),r_(i−1,j+1), r_(i,j+1), r_(i+1,j+1), r_(i+1,j), r_(i+1,j−1), andr_(i,j−1) of corresponding pixels of S to locations 880, 882, 884, 888,890, and 892, respectively. Further, reconstruction component 710 cancopy green and blue subpixels of the corresponding pixels of S tolocations of L′ in a manner similar to the description above. As such,reconstruction component 710 can generate neighboring red, green, andblue components (or subpixels) using a directional weighted average ofrespective neighboring components as described above to reconstructvirtual image L′.

FIG. 9 illustrates a minimum mean square error (MMSE) subpixel-baseddown-sampling (MMSE-SD) system 900, in accordance with an embodiment.MMSE-SD system 900 includes MMSE-SD component 910 that can be configuredto determine an optimal low resolution image S. MMSE-SD component 910can receive information associated with high resolution image L,including 3M×3N color components, or subpixels, R, G, and B of highresolution image L. Further, MMSE-SD can generate 3M×3N colorcomponents, or subpixels, R′, G′, and B′ of L′ based on reconstructionfunction ƒ_(r)(r) associated with Equations 1-4 and 13, reconstructionfunction ƒ_(g)(g) associated with Equations 5-8 and 13, andreconstruction ƒ_(b)(b) associated with Equations 9-12 and 13,respectively.

In an aspect, MMSE-SD component 910 can separately derive subpixels r,g, and b of low resolution image S by minimizing a mean square error(MSE) between L and L′, according to Equation 13 as follows:

$\begin{matrix}{{{{\min\limits_{r,g,b}{{R - R^{\prime}}}_{2}^{2}} + {{G - G^{\prime}}}_{2}^{2} + {{B - B^{\prime}}}_{2\;}^{2}}{s.t.\mspace{14mu} R^{\prime}} = {{fr}(r)}}\mspace{45mu} {G^{\prime} = {{fg}(g)}}\mspace{45mu} {B^{\prime} = {{{fb}(b)}.}}} & (13)\end{matrix}$

As such, MMSE-SD component 910 can derive subpixel r of low resolutionimage S according to Equations 14-16 as follows:

$\begin{matrix}{{\min\limits_{r}{{R - R^{\prime}}}_{2}^{2}}{{{s.t.\mspace{14mu} R^{\prime}} = {{fr}(r)}},}} & (14) \\{{{H_{r}r} = {H_{R}R}},} & (15) \\{{r = {{\left( {H_{r}^{- 1}H_{R}} \right)R} = {HR}}},} & (16)\end{matrix}$

in which H_(r) and H_(R) are block-circulant matrices of size MN×MN andMN×9MN, respectively, according to Equations 17-26 as follows; in whichR is a row-ordered vector of size 9MN×1 from the red component of L, andr is the row-ordered vector of size MN×1 from the red component of S,respectively, and in which H=Hr⁻¹H_(R) is a block circulant matrix ofsize MN×9MN, including blocks of size N×9N that are block-tri-circulant(see below). Each of the N×9N sized blocks include three sub-blocks ofsize N×3N, and each of the three sub-blocks is block-circulant (seebelow) with a block size of 1×3:

$\begin{matrix}{{H_{r} = {\begin{matrix}A & B_{2} & 0 & \ldots & B_{1} \\B_{1} & A & B_{2} & \ldots & 0 \\0 & B_{1} & A & B_{2} & \ldots \\\vdots & \ddots & \ddots & \ddots & \vdots \\B_{2} & 0 & \ldots & B_{1} & A\end{matrix}}},} & (17) \\{{H_{R} = {\begin{matrix}C & D_{2} & E_{2} & 0 & 0 & 0 & 0 & \ldots & 0 & E_{1\;} & D_{1} \\0 & E_{1} & D_{1} & C & {D_{2}\;} & E_{2} & 0 & 0 & 0 & \ldots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots \\0 & \ldots & 0 & E_{1} & D_{1} & C & D_{2} & E_{2} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & \ldots & 0 & E_{1} & D_{1} & C & D_{2} & E_{2}\end{matrix}}},} & (18) \\{{A = {\begin{matrix}k_{0} & k_{1} & 0 & \ldots & k_{1} \\k_{1} & k_{0} & k_{1} & 0 & \ldots \\\vdots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & k_{1} & k_{0} & k_{1} \\k_{1} & 0 & \ldots & k_{1} & k_{0}\end{matrix}}},} & (19) \\{{B_{1} = {\begin{matrix}k_{2} & k_{4} & 0 & \ldots & k_{3} \\k_{3} & k_{2} & k_{4} & 0 & \ldots \\\vdots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & k_{3} & k_{2} & k_{4} \\k_{4} & 0 & \ldots & k_{3} & k_{2}\end{matrix}}},} & (20) \\{{B_{2} = {\begin{matrix}k_{2} & k_{3} & 0 & \ldots & k_{4} \\k_{4} & k_{2} & k_{3} & 0 & \ldots \\\vdots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & k_{4} & k_{2} & k_{3} \\k_{3} & 0 & \ldots & k_{4} & k_{2}\end{matrix}}},} & (21)\end{matrix}$

in which k₀=1+2α₁ ²+2β₁ ²+2α₂ ²+2β₂ ²+2α₃ ²+2β₃ ²+2α₄ ²+2β₄ ², k₁=2α₁β₁,k₂=2α₂β₂, k₃=2α₃β₃, and k₄=2α₄β₄.

H_(R) includes M×3M blocks (e.g., either C, D₁, D₂, E₁, E₂, or 0),wherein each block of the M×3M blocks is a matrix of size N×3N asdescribed by Equations 22-26 as follows:

$\begin{matrix}{{C = {\begin{matrix}1 & \alpha_{1} & \beta_{1} & 0 & 0 & \ldots & \beta_{1} & \alpha_{1} \\0 & \beta_{1} & \alpha_{1} & 1 & \alpha_{1} & \beta_{1} & \ldots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & \beta_{1} & \alpha_{1} & 1 & \alpha_{1} & \beta_{1}\end{matrix}}},} & (22) \\{{D_{1} = {\begin{matrix}\alpha_{2} & \alpha_{4} & 0 & 0 & 0 & \ldots & 0 & \alpha_{3} \\0 & 0 & \alpha_{3} & \alpha_{2} & \alpha_{4} & 0 & \ldots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & 0 & \alpha_{3} & \alpha_{2} & \alpha_{4} & 0\end{matrix}}},} & (23) \\{{D_{2} = {\begin{matrix}\alpha_{2} & \alpha_{3} & 0 & 0 & 0 & \ldots & 0 & \alpha_{4} \\0 & 0 & \alpha_{4} & \alpha_{2} & \alpha_{3} & 0 & \ldots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & 0 & \alpha_{4} & \alpha_{2} & \alpha_{3} & 0\end{matrix}}},} & (24) \\{{E_{1} = {\begin{matrix}\beta_{2} & 0 & \beta_{4} & 0 & \ldots & 0 & \beta_{3} & 0 \\0 & \beta_{3} & 0 & \beta_{2} & 0 & \beta_{4} & \ldots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & \beta_{3} & 0 & \beta_{2} & 0 & {\beta_{4}\;}\end{matrix}}},} & (25) \\{E_{2} = {{\begin{matrix}\beta_{2} & 0 & \beta_{3} & 0 & \ldots & 0 & \beta_{4} & 0 \\0 & \beta_{4} & 0 & \beta_{2} & 0 & \beta_{3} & \ldots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & \beta_{4} & 0 & \beta_{2} & 0 & \beta_{3\;}\end{matrix}}.}} & (26)\end{matrix}$

Consider 3 elements in the first row of C as a block, such as [1 α₁ β₁].Such a block is repeatedly shifted to the right by 3 positions in thesubsequent rows. Thus, each of C, D₁, D₂, E₁, E₂, or 0 isblock-circulant. Similarly, considering 3 sub-blocks of H_(R) in thehorizontal direction (e.g. [C D₂ E₂]) as a block, which appears in thefirst row in H_(R) and is repeatedly shifted to the right by 3 blockpositions in the subsequent rows, H_(R) is a “block-circulant” matrix.

Here, we call a matrix of size N×9N “block-tri-circulant” if it containsthree sub-blocks of size N×3N, each of which is block-circulant (e.g. [CD₂ E₂] is block-tri-circulant). Therefore, H_(R) is a block circulantmatrix, with blocks of size N×9N that are block-tri-circulant.

As described above, H is a block circulant matrix of size MN×9MN,including N×9N blocks that are block-tri-circulant. Each of the N×9Nblocks include three sub-blocks of size N×3N, and each of the threesub-blocks is block-circulant with a block size of 1×3. In other words,each row of a (k,l)^(th) sub-block, k=1, . . . , M, and l=1, . . . , 3M,of the N×9 N blocks of block circulant matrix H has 3N coefficients andis equal to the previous row rotated, or shifted, to the right by 3sub-block positions. The m^(th) row of the (k,l)^(th) sub-block of H isassociated with an inner-product with the l^(th) row of L and adds aterm to the m^(th) element of the k^(th) row of S, i.e., the (k,m)^(th)pixel of S.

MMSE-SD component 910 can be configured to generate a 3N×3N circulantmatrix using the first row of the (k,l)'^(h) sub-block, as the row has3N coefficients and is equal to the previous row rotated to the right by3 sub-block positions. Further, MMSE-SD component 910 can perform a 3:1down-sampling on the 3N×3N circulant matrix in a vertical directionutilizing Equation 27 described below:

$\begin{matrix}{{I_{a}\left( {i,j} \right)} = \left\{ {\begin{matrix}{1,} & {{i = 1},\ldots \mspace{14mu},N,{{j = {{3i} - 2}};}} \\{0,} & {{otherwise}.}\end{matrix}.} \right.} & (27)\end{matrix}$

Thus, MMSE-SD component 910 can be configured to (1) perform a 1−Dconvolution of the first row of the (k,l)^(th) sub-block of blockcirculant matrix H with a periodic extension of the l^(th) row of L togenerate a row of size 3N; (2) perform a 3:1 down-sampling of the row ofsize 3N in a horizontal direction; and (3) add a result of the 3:1down-sampling to the k^(th) row of S.

In an aspect, MMSE-SD component 910 can perform such operations of firstrows of sub-blocks of block circulant matrix H on L (with k=1, l=1, . .. , 3M), effectively applying a 2-D spatial-invariant linear filter on aperiodic extension of L to obtain a row of size 3N, in which MMSE-SD canobtain coefficients of the 2-D spatial-invariant filter from the firstrow of block circulant matrix H. Further, MMSE-SD component 910 can 3:1down-sample the row in a horizontal direction to obtain the 1^(st) rowof S.

As stated above, H is a block-circulant matrix including N×9N blocksthat are block-tri-circulant (each block including three sub-blocks ofsize N×3N that are block-circulant). Further, each row of the N×9Nblocks can be considered as a row of sub-blocks of size N×3N, in whicheach row of sub-blocks is equivalent to a previous row rotated to theright by 3 sub-block positions. As such, in an embodiment, MMSE-SDcomponent 910 can be configured to generate a derived block-circulantmatrix of size 3MN×9MN that includes blocks of size N×3N. Further,MMSE-SD component 910 can be configured to compute H as the product of amatrix I_(b) (defined by Equation 28 below) of size MN×3MN and thederived block-circulant matrix:

$\begin{matrix}{{I_{b}\left( {i,j} \right)} = \left\{ {\begin{matrix}{I_{N},} & {{i = 1},\ldots \mspace{14mu},M,{{j = {{3i} - 2}};}} \\{0,} & {{otherwise}.}\end{matrix}.} \right.} & (27)\end{matrix}$

In one aspect, the product of matrix I_(b) and the derivedblock-circulant matrix is a 3:1 down-sampling on the sub-blocks of thederived block-circulant matrix in a vertical direction. As such, MMSE-SDcomponent 910 applies a 2-D spatial-invariant filter, of size 3M×3N, onthe periodic extension of L to obtain an image of size 3N×3N. Such afilter can be independent of high resolution image (L), and thuspre-computed and stored, e.g., in a storage medium. In an aspect, the2-D spatial-invariant filter can be set to a size k×k, e.g., k=15.Further MMSE-SD component 910 can 3:1 down-sample the image in ahorizontal direction and 3:1 down-sample the image in a verticaldirection.

Now referring to FIG. 10, a block diagram of a down-sampling environment1000 including a low resolution display 1010 is illustrated, inaccordance with an embodiment. Down-sampling environment 1000 caninclude system 100, which can receive high resolution image (L)associated with, e.g., HDTV information, HD movies, etc. System 100 caninclude display interface component (not shown), that can couple todisplay 1010 to display a low resolution image (S) including subpixelssampled and/or generated, e.g., via sampling component 110 of system100, from high resolution image (L). Display 1010 can include a lowerresolution display, e.g., included in a handheld device such as acellular phone, a PMP, a PDA, etc.

FIGS. 11-12 illustrate methodologies in accordance with the disclosedsubject matter. For simplicity of explanation, the methodologies aredepicted and described as a series of acts. It is to be understood andappreciated that the subject innovation is not limited by the actsillustrated and/or by the order of acts. For example, acts can occur invarious orders and/or concurrently, and with other acts not presented ordescribed herein. Furthermore, not all illustrated acts may be requiredto implement the methodologies in accordance with the disclosed subjectmatter. In addition, those skilled in the art will understand andappreciate that the methodologies could alternatively be represented asa series of interrelated states via a state diagram or events.Additionally, it should be further appreciated that the methodologiesdisclosed hereinafter and throughout this specification are capable ofbeing stored on an article of manufacture to facilitate transporting andtransferring such methodologies to computers. The term article ofmanufacture, as used herein, is intended to encompass a computer programaccessible from any computer-readable device, carrier, or media.

Referring now to FIG. 11, a process 1100 associated with minimum meansquare error subpixel-based down-sampling (MMSE-SD) is illustrated, inaccordance with an embodiment. At 1110, a high resolution image can bedivided into at least two blocks, wherein each block of the at least twoblocks includes a 3×3 array of pixels. Red, green, and blue subpixelscan be diagonally selected from respective adjacent pixels of each blockof the at least two blocks at 1120. At 1130, a low resolution,down-sampled image S can be created based on subpixels selected at 1120.

FIG. 12 illustrates a process 1200 for creating an optimal lowresolution image (S), in accordance with an embodiment. At 1210, avirtual image can be derived, based on the low resolution image createdat 1120 (see FIG. 11). Mean square error between the virtual image andthe low resolution image can be minimized at 1220. At 1230, the optimallow resolution image (S) can be created, generated, etc. based on aresult of step 1220.

As it employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to comprising, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsand/or processes described herein. Processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of mobile devices. A processor may also beimplemented as a combination of computing processing units.

In the subject specification, terms such as “store,” “data store,” “datastorage,” “database,” “storage medium,” and substantially any otherinformation storage component relevant to operation and functionality ofa component and/or process, refer to “memory components,” or entitiesembodied in a “memory,” or components comprising the memory. It will beappreciated that the memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory.

By way of illustration, and not limitation, nonvolatile memory, forexample, can be included in storage systems described above,non-volatile memory 1322 (see below), disk storage 1324 (see below), andmemory storage 1346 (see below). Further, nonvolatile memory can beincluded in read only memory (ROM), programmable ROM (PROM),electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), or flash memory. Volatile memory can include random accessmemory (RAM), which acts as external cache memory. By way ofillustration and not limitation, RAM is available in many forms such assynchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM),double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), SynchlinkDRAM (SLDRAM), and direct Rambus RAM (DRRAM). Additionally, thedisclosed memory components of systems or methods herein are intended tocomprise, without being limited to comprising, these and any othersuitable types of memory.

In order to provide a context for the various aspects of the disclosedsubject matter, FIG. 13, and the following discussion, are intended toprovide a brief, general description of a suitable environment in whichthe various aspects of the disclosed subject matter can be implemented,e.g., various processes associated with FIGS. 1-12. While the subjectmatter has been described above in the general context ofcomputer-executable instructions of a computer program that runs on acomputer and/or computers, those skilled in the art will recognize thatthe subject innovation also can be implemented in combination with otherprogram modules. Generally, program modules include routines, programs,components, data structures, etc. that perform particular tasks and/orimplement particular abstract data types.

Moreover, those skilled in the art will appreciate that the inventivesystems can be practiced with other computer system configurations,including single-processor or multiprocessor computer systems,mini-computing devices, mainframe computers, as well as personalcomputers, hand-held computing devices (e.g., PDA, phone, watch),microprocessor-based or programmable consumer or industrial electronics,and the like. The illustrated aspects can also be practiced indistributed computing environments where tasks are performed by remoteprocessing devices that are linked through a communications network;however, some if not all aspects of the subject disclosure can bepracticed on stand-alone computers. In a distributed computingenvironment, program modules can be located in both local and remotememory storage devices.

With reference to FIG. 13, a block diagram of a computing system 1300operable to execute the disclosed systems and methods is illustrated, inaccordance with an embodiment. Computer 1312 includes a processing unit1314, a system memory 1316, and a system bus 1318. System bus 1318couples system components including, but not limited to, system memory1316 to processing unit 1314. Processing unit 1314 can be any of variousavailable processors. Dual microprocessors and other multiprocessorarchitectures also can be employed as processing unit 1314.

System bus 1318 can be any of several types of bus structure(s)including a memory bus or a memory controller, a peripheral bus or anexternal bus, and/or a local bus using any variety of available busarchitectures including, but not limited to, Industrial StandardArchitecture (ISA), Micro-Channel Architecture (MSA), Extended ISA(EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB),Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus(USB), Advanced Graphics Port (AGP), Personal Computer Memory CardInternational Association bus (PCMCIA), Firewire (IEEE 1194), and SmallComputer Systems Interface (SCSI).

System memory 1316 includes volatile memory 1320 and nonvolatile memory1322. A basic input/output system (BIOS), containing routines totransfer information between elements within computer 1312, such asduring start-up, can be stored in nonvolatile memory 1322. By way ofillustration, and not limitation, nonvolatile memory 1322 can includeROM, PROM, EPROM, EEPROM, or flash memory. Volatile memory 1320 includesRAM, which acts as external cache memory. By way of illustration and notlimitation, RAM is available in many forms such as SRAM, dynamic RAM(DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM),enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), Rambus direct RAM(RDRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM(RDRAM).

Computer 1312 can also include removable/non-removable,volatile/non-volatile computer storage media, networked attached storage(NAS), e.g., SAN storage, etc. FIG. 13 illustrates, for example, diskstorage 1324. Disk storage 1324 includes, but is not limited to, deviceslike a magnetic disk drive, floppy disk drive, tape drive, Jaz drive,Zip drive, LS-100 drive, flash memory card, or memory stick. Inaddition, disk storage 1324 can include storage media separately or incombination with other storage media including, but not limited to, anoptical disk drive such as a compact disk ROM device (CD-ROM), CDrecordable drive (CD-R Drive), CD rewritable drive (CD-RW Drive) or adigital versatile disk ROM drive (DVD-ROM). To facilitate connection ofthe disk storage devices 1324 to system bus 1318, a removable ornon-removable interface is typically used, such as interface 1326.

It is to be appreciated that FIG. 13 describes software that acts as anintermediary between users and computer resources described in suitableoperating environment 1300. Such software includes an operating system1328. Operating system 1328, which can be stored on disk storage 1324,acts to control and allocate resources of computer 1312. Systemapplications 1330 take advantage of the management of resources byoperating system 1328 through program modules 1332 and program data 1334stored either in system memory 1316 or on disk storage 1324. It is to beappreciated that the disclosed subject matter can be implemented withvarious operating systems or combinations of operating systems.

A user can enter commands or information into computer 1312 throughinput device(s) 1336. Input devices 1336 include, but are not limitedto, a pointing device such as a mouse, trackball, stylus, touch pad,keyboard, microphone, joystick, game pad, satellite dish, scanner, TVtuner card, digital camera, digital video camera, web camera, and thelike. These and other input devices connect to processing unit 1314through system bus 1318 via interface port(s) 1338. Interface port(s)1338 include, for example, a serial port, a parallel port, a game port,and a universal serial bus (USB). Output device(s) 1340 use some of thesame type of ports as input device(s) 1336.

Thus, for example, a USB port can be used to provide input to computer1312 and to output information from computer 1312 to an output device1340. Output adapter 1342 is provided to illustrate that there are someoutput devices 1340 like monitors, speakers, and printers, among otheroutput devices 1340, which use special adapters. Output adapters 1342include, by way of illustration and not limitation, video and soundcards that provide means of connection between output device 1340 andsystem bus 1318. It should be noted that other devices and/or systems ofdevices provide both input and output capabilities such as remotecomputer(s) 1344.

Computer 1312 can operate in a networked environment using logicalconnections to one or more remote computers, such as remote computer(s)1344. Remote computer(s) 1344 can be a personal computer, a server, arouter, a network PC, a workstation, a microprocessor based appliance, apeer device, or other common network node and the like, and typicallyincludes many or all of the elements described relative to computer1312.

For purposes of brevity, only a memory storage device 1346 isillustrated with remote computer(s) 1344. Remote computer(s) 1344 islogically connected to computer 1312 through a network interface 1348and then physically connected via communication connection 1350. Networkinterface 1348 encompasses wire and/or wireless communication networkssuch as local-area networks (LAN) and wide-area networks (WAN). LANtechnologies include Fiber Distributed Data Interface (FDDI), CopperDistributed Data Interface (CDDI), Ethernet, Token Ring and the like.WAN technologies include, but are not limited to, point-to-point links,circuit switching networks like Integrated Services Digital Networks(ISDN) and variations thereon, packet switching networks, and DigitalSubscriber Lines (DSL).

Communication connection(s) 1350 refer(s) to hardware/software employedto connect network interface 1348 to bus 1318. While communicationconnection 1350 is shown for illustrative clarity inside computer 1312,it can also be external to computer 1312. The hardware/software forconnection to network interface 1348 can include, for example, internaland external technologies such as modems, including regular telephonegrade modems, cable modems and DSL modems, ISDN adapters, and Ethernetcards.

The above description of illustrated embodiments of the subjectdisclosure, including what is described in the Abstract, is not intendedto be exhaustive or to limit the disclosed embodiments to the preciseforms disclosed. While specific embodiments and examples are describedherein for illustrative purposes, various modifications are possiblethat are considered within the scope of such embodiments and examples,as those skilled in the relevant art can recognize.

In this regard, while the disclosed subject matter has been described inconnection with various embodiments and corresponding Figures, whereapplicable, it is to be understood that other similar embodiments can beused or modifications and additions can be made to the describedembodiments for performing the same, similar, alternative, or substitutefunction of the disclosed subject matter without deviating therefrom.Therefore, the disclosed subject matter should not be limited to anysingle embodiment described herein, but rather should be construed inbreadth and scope in accordance with the appended claims below.

1. A method comprising: partitioning a two-dimensional (2-D) array ofpixels into 2-D blocks of pixels; alternately sampling subpixels of ablock of pixels of the 2-D blocks of pixels in a diagonal direction; andgenerating an image based on a result of the alternately sampling thesubpixels of the block of pixels.
 2. The method of claim 1, wherein thepartitioning the 2-D array of pixels includes partitioning the 2-D arrayof pixels into 3×3 arrays of pixels.
 3. The method of claim 1, whereinthe alternately sampling subpixels of the block of pixels includesalternately selecting a red subpixel, a green subpixel, and a bluesubpixel from consecutive pixels of the block of pixels in the diagonaldirection.
 4. The method of claim 3, wherein the alternately selectingincludes alternately selecting the red subpixel, the green subpixel, andthe blue subpixel from consecutive pixels of a 3×3 array of pixels inthe diagonal direction.
 5. The method of claim 1, further comprising:deriving, based on the image, a virtual image according to a size of another image associated with the 2-D array of pixels.
 6. The method ofclaim 5, wherein the generating the image includes: minimizing a meansquare error between the virtual image and the other image.
 7. Themethod of claim 1, wherein the generating the image includes determiningat least one color component of a low resolution image associated withthe result based on a block circulant matrix.
 8. The method of claim 7,wherein the determining the at least one color component includesdetermining the at least on color component based on a block circulantmatrix of size MN×9MN including N×9N arrays of blocks that areblock-tri-circulant.
 9. A system comprising: a partition componentconfigured to: receive a first array of pixels; and divide the firstarray of pixels into two-dimensional (2-D) blocks of pixels; and asampling component configured to: diagonally down-sample subpixels of ablock of the 2-D blocks; and generate a second array of pixels based onthe down-sampled subpixels.
 10. The system of claim 9, wherein thesampling component is further configured to: alternately sample a redsubpixel, a green subpixel, and a blue subpixel of adjacent pixels ofthe block in a diagonal direction; and generate the second array ofpixels based on the red subpixel, the green subpixel, and the bluesubpixel.
 11. The system of claim 9, wherein the block is a 3×3 array ofpixels, and wherein the sampling component is further configured toalternately select subpixels of the 3×3 array of pixels in a diagonaldirection.
 12. The system of claim 9, further comprising: areconstruction component configured to create a virtual image based on,at least in part, the second array of pixels.
 13. The system of claim12, wherein the reconstruction component is further configured to createthe virtual image utilizing a directional weighted average ofneighboring subpixels of the second array of pixels.
 14. The system ofclaim 12, further comprising: a minimum mean square error (MMSE)subpixel-based down-sampling (MMSE-SD) component configured to determinean optimal low resolution image based on, at least in part, respectivecolor components of the virtual image and a high resolution imageassociated with the first array of pixels.
 15. The system of claim 14,wherein the MMSE-SD component is further configured to: minimize a meansquare error between the virtual image and the high resolution image;and determine the optimal low resolution image based on the mean squareerror.
 16. The system of claim 14, wherein the MMSE-SD component isfurther configured to determine color components of the optimal lowresolution image according to a block circulant matrix.
 17. The systemof claim 17, wherein the block circulant matrix is a MN×9MN matrixincluding N×9N arrays of blocks that are block-tri-circulant.
 18. Anapparatus, comprising: means for dividing a high resolution image intoblocks of pixels; means for selecting subpixels of a block of the blocksalong a diagonal direction; and means for creating a low resolutionimage based on the selected subpixels.
 19. The apparatus of claim 18,further comprising: means for creating a virtual high resolution imageusing the low resolution image.
 20. The apparatus of claim 19, furthercomprising: means for minimizing a mean square error between the highresolution image and the virtual image; and means for determining anoptimal low resolution image based on an output of the means forminimizing the mean square error.